Can you elaborate what you mean by "not on the support of the classes you are trying to distinguish"? In the case of your digit classifier, in order to improve your classifier, you want to train with an extra output that says "not a digit"?
Yes that is the idea of BadGAN, which had a long run of being SoA for semisupervised learning (SSL).
Instead of fake/real you have fake/0/1/2/3/4/5/6/7/8/9. In the setting for this article that would mean to distinguish between real news and fake news your generator would learn to generate news that is neither, whatever that means. But BadGAN works really well for images to improve your classifier, not sure if anyone has used it elsewhere. Maybe for text SSL isn't that important.
Really for NN generated text in the wild now that I think of it you'd probably want a more standard technique like looking at the metadata. Or you could form some knowledge graph tying together nouns and just say anything that hits too many unconnected topics is NN generated.
